Characterizations of generalized entropy functions by functional equations

We shall show that a two-parameter extended entropy function is characterized by a functional equation. As a corollary of this result, we obtain that the Tsallis entropy function is characterized by a functional equation, which is a different form used in \cite{ST} i.e., in Proposition \ref{prop01} in the present paper. We also give an interpretation of the functional equation giving the Tsallis entropy function, in the relation with two non-additive properties.


Introduction
Recently, generalized entropies have been studied from the mathematical point of view. The typical generalizations of Shannon entropy [2] are Rényi entropy [3] and Tsallis entropy [4]. The Rényi entropy and the Tsallis entropy are defined by R q (X) = 1 1 − q log n j=1 x q j , (q = 1, q > 0) and S q (X) = n j=1 x j − x q j q − 1 , (q = 1, q > 0) for a given information source X = {x 1 , · · · , x n } with the probability p j ≡ P r(X = x j ). Both entropies recover Shannon entropy: p j log p j in the limit q → 1. The uniqueness theorem for the Tsallis entropy was firstly given in [5] and improved in [6]. Throughout this paper, a parametric extended entropy such as Rényi entropy and Tsallis entropy and so on, is called by a generalized entropy.
We note that the Rényi entropy has the additivity: but the Tsallis entropy has the non-additivity: where X × Y means that X and Y are independent random variables. Therefore we have a definitive difference for these entropies, although we have the simple relation between them: The Tsallis entropy is rewritten by where the q-logarithmic function is defined by which uniformly converges to log x in the limit q → 1. Since Shannon entropy can be regarded as the expectation value for each value − log p j , we may consider that the Tsallis entropy can be regarded as the q-expectation value for each value − ln q p j , as an anology to Shannon entropy. Where q-expectation value E q is defined by However, the q-expectation value E q lacks the fundamental property such as E(1) = 1, so that it was considered to be inadequate to adopt as a generalized definition of the usual expectation value. Then the noremalized q-expectation value was introduced: and by using this, the normalized Tsallis entropy was defined by We easily find that we have the following non-additivity relation for the normalized Tsallis entropy: As for the details on the mathematical properties of the normalized Tsallis entropy, see [7] for example. The difference between two non-additivities Eq.(1) and Eq.(3) is the signature of the coefficient 1 − q in the third term of the right hand sides. We note that the Tsallis entropy is also rewritten by so that we may regard it as the expectation value such as S q (X) = E 1 ln q 1 p j , where E 1 means the usual expectation value. However, if we adopt this formulation in the definition of the Tsallis conditional entropy, we do not have an important property such as a chain rule. (See [8] for details.) Therefore we often adopt the formulation using the q-expectation value.
As a further generalization, for two positive numbers α and β, a two-parameter extended entropy: has been studied in many literatures [9,10,11,12,13,14,15]. In the paper [1], a characterization of the Tsallis entropy function was proven by using the functional equation. In this paper, we shall show that the two-parameter extended entropy function is characterized by the functional equation.

A review of the characterization of Tsallis entropy function by the functional equation
The following proposition was originally given in [1] by the simple and elegant proof. Here we give the alternative proof along to the proof given in [16].
If the differentiable nonnegative function f q with positive parameter q ∈ R satisfies the following functional equation: then the function f q is uniquely given by where c q is a nonnegative constant depending only on the parameter q.

Proof:
If we put y = 1 in Eq. (5), then we have f q (1) = 0. From here we assume y = 1. We also put Putting x = 1 2 in (6), we have Substituting y 2 into y, we have By repeating similar substitutions, we have g q y 2 N = g q 1 2 y q−1 1 + 1 2 due to q > 0. Differentiating Eq.(6) by y, we have x)) y q−2 Putting y = 1 in the above equation, we have where c q = −g ′ q (1). By integrating the equation (6) from 2 −N to 1 with respect to y and performing the conversion of the variables, we have By differentiating the above equation with respect to x, we have Taking the limit N → ∞ in the above, we have thanks to (7). From the equations (8) and (10), we have the following differential equation: xg ′ q (x) + (1 − q)g q (x) = −c q . This differential equation has the following general solution: where d q is a integral constant depending on q. From g q (1) = 0, we have d q = cq 1−q . Thus we have Finally we have From f q (x) ≥ 0, we have c q ≥ 0.
If we take the limit as q → 1 in Theorem 2.1, we have the following corollary.

Corollary 2.2 ([16])
If the differentiable nonnegative function f satisfies the following functional equation: then the function f is uniquely given by where c is a nonnegative constant.

Main results
In this section, we give a characterization of a two-parameter extended entropy function by the functional equation. Before we give our main theorem, we review the following result given by Pl.Kannappan [17,18].
In the following theorem, we adopt a simpler condition than Eq.(12).
Theorem 3.2 If the differentiable nonnegative function f α,β with two positive parameters α, β ∈ R satisfies the following functional equation: then the function f α,β is uniquely given by where c α,β and c α are nonnegative constants depending only on the parameters α (and β).
x β+1 x −β f α,β (x) That is, we have, Integrating both sides on the above equation with respect to x, we have where d α,β is a integral constant depending on α and β. Therefore we have Also by f α,β (x) ≥ 0, we have c α,β ≥ 0.
As for the case of α = β, we can prove by the similar way.

Corollary 3.3
If the differentiable nonnegative function f q with a positive parameter q ∈ R satisfies the following functional equation: then the function f q is uniquely given by where c q is a nonnegative constant depending only on the parameter q.
Here we give an interpretation of the functional equation (16) from the view of Tsallis statistics.
Then the functional equation (17) can be regarded as the sum of two following functional equations: Two equations (18) and (19) imply the following equations for i = 1, · · · , n and j = 1, · · · , m.
Taking the sum for Eq.(20) and Eq.(21) on i and j with simple calculations, we have two nonadditivity relations given in Eq.(1) and Eq. (3), where we put c q = 1 for a simplicity. Therefore we can conclude that two functional equations (18) and (19), which are the essential parts of the non-additivity relations Eq.(1) and Eq. (3), characterize the Tsallis entropy function. In other words, the Tsallis entropy function can be characterized by two non-additivity relations Eq.(1) and Eq. (3).
If we again take the limit as q → 1 in Corollary 3.3, we have the following corollary.
Corollary 3.5 If the differentiable nonnegative function f satisfies the following functional equation: f (xy) = yf (x) + xf (y), (0 < x, y ≤ 1) then the function f is uniquely given by where c is a nonnegative constant.